ITT: People who have never watched the old series.
Here's the problem with the new series, pacing. The old show had subtlety, and wasn't always just the doctor rushing around trying to save everyone. Most of the time it was just him trying not to get his geriatric ass killed. With the new series it's all, "I am THE DOCTOR! Fear me bad guys! I might look like a dork but I am really legendary warrior! Rawr!"
In the old series he's basically just a dude who stole a "car" with his granddaughter. And is on the run from the feds. And each doctor brought something unique to it. Yes, the special effects were ass but it was 50 years ago for christ's sake. Cut em some slack.
To my mind the place to begin is unquestionably in the beginning. An Unearthly Child. Ian, Barbara and Susan, with one of the crotchetiest crotchety old men you will ever meet playing the doctor. Not all the episodes are great, but many of them are. And it all helps to weave together not just a story, but a mythology.
Consider the parallel. In the new series you have the doctor saving the entire frigging universe in nearly EVER episode, or at the very least a planet. In the old series you have the doctor doing such heroic things as looking for mercury on an undeveloped world! Trying to get back to the tardis to escape! Trying to find missing companions! Or just exploring! Just a "Hey, we're here, but I have no idea where here is and I'm reasonably sure the atmosphere won't kill us the second we open the door. LET'S GO ON AN ADVENTURE!" Pacing again. If you have the entire space time continuum in peril every episode, what's left to risk? Hell, some of my favorite episodes are the ones where the entire thing takes place on the tardis.
And that notion of pacing seems to be one they've forgotten in the new series. Consider EVERY season finale of the last decade or so. Nearly universally they are all almost absurdly over the top. The doctor being saved by people wishing hard enough... Yeah. Full on tinkerbell. I get it, but there's such a thing as subtlety too. Even the master as a villain went from a devious, sociopathic intellectual to cartoon supervillan.
It's not that the new series is bad, it certainly beats the alternative, but something's been lost in translation between 1962 and now. And to me that's a damned tragedy.
If you're looking for a quick and easy jumping on point, and you don't have the patience to sit through the old black and white Hartnells and Troughtons, Start with Jon Pertwee or Tom Baker (Still the quintessential doctor, though Tennant was close). You'll get the advantage of seeing what the series is capable of when they're not competing for ratings.